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The shape of the response curve is also surprising. Figure 3 shows normalized step-response curves for 1, 5 and 100 equal interacting lags. Note first that all curves cross t/Στ = 1 at 63.2% response. Then observe how little separates the curves as n increases. Raising n above 100 reveals no discernible change whatsoever, either to the eye or to the controller closing the loop around the process. This has an important corollary: simulation using as few as 20 interacting lags is quite sufficient to represent any number, including infinity, i.e., an infinite number of infinitesimally small lags. The latter is representative of a truly distributed process, comprised not of a series of discrete stages but of a continuum of capacity and resistance.
To estimate optimum controller settings requires some knowledge of process parameters, obtainable through design information or dynamic testing or both. Most tuning procedures are based on open-loop information, which includes estimated process gain, dead time and time constant—three process parameters. The most prominent pioneers in this field, John Ziegler and Nathan Nichols, estimated only two properties of an open-loop step-response (See “Optimum Settings for Automatic Controllers,” Trans. ASME, Nov. 1942) as described in Figure 4. Realizing that many slow processes would require too much time to reach steady state following a step upset, they elected to omit an estimate of steady-state gain. This makes sense, in that reaching within 1% of the steady state for a distributed lag requires an elapsed time of 4Στ, which for a 100-tray column would be 28 hours. Management is unlikely to accept such long intervals of open-loop operation, and other disturbances are likely to develop during the wait, as well.
Analyzing the step response
Instead, they approximated the response curve with two straight lines as in Figure 4, one a tangent to the curve at its steepest slope, and the other, a zero-response line ending at the zero-intercept of the tangent. The time at which the intercept is reached is considered estimated dead time, τde. Ziegler and Nichols then measured the slope of the tangent in terms of percent-per-minute, and compared it with the size of the step input. Their method can be simplified somewhat by marking the size of the manipulated-variable step Δm in percent of scale against the tangent, also in percent of scale. In the example of Figure 4, the process steady-state gain exceeds unity, in that the net change in the controlled variable exceeds Δm, but in some cases it could be less than unity. The time required for the controlled variable to respond an amount equal to Δm if it were to follow the tangent is its estimated time constant τ1e.
This method is applicable only theoretically to a non-self-regulating (integrating) process, where the controlled variable would continue to follow the tangent indefinitely. But it turns out to be a very effective method for estimating the parameters of a distributed lag because of the singularity of its response curve. As it happens, the two parameters needed to define the distributed process, steady-state gain, Kp, and response time, Στ, are directly related to the features of the response curve as estimated above:
Ziegler and Nichols found that closed-loop testing under proportional control can produce still more-accurate estimates of controller settings. In this method, they reduced the controller to effectively proportional-only action by setting derivative time to zero and integral time to maximum. They then reduced the proportional band incrementally until a uniform oscillation was produced. From this simple test they obtained two pieces of information: the proportional band that produced the sustained oscillation and the period of the oscillation.
Under these conditions, a distributed process will oscillate uniformly with the proportional band in percent set at 8.5Kp, and the resulting period of oscillation is observed to be 0.643Στ. The process parameters are then:
where Pu is the undamped proportional band and τu is the period of the undamped cycle.
It is often possible to calculate these parameters from known process information, when available, which can avoid testing, and even give valuable insight into the potential for parameter variations. For example, consider a stirred tank used for blending ingredients to a controlled composition. It can be modeled as a distributed lag whose response time Στ equals the residence time of the vessel, V/F, volume divided by flow rate. Steady-state gain, Kp, is the product of valve, process and transmitter gains. As an example, consider temperature control of an air-conditioned space against an ambient of 90 °F. If it is known that maximum chilling reduces the room temperature to 60 °F, and the range of the temperature transmitter is 50 °F to 100 °F, then Kp = (90 – 60)/(100 – 50) = 0.60.
A process with a fixed volume, V, will have a residence time that varies inversely with flow, F, through it. This is the case for the stirred tank described above. However, its steady-state gain also varies with flow in the same proportion. Consider, for example, an incremental flow of brine added to a given flow of water to produce a measured increase in salt concentration of the blend. If the water flow is then reduced by half, the same step change in brine flow will produce twice the change in salt concentration as before. Because the gain- and residence-time parameters change together, there is no net change in the slope of the response curve at its steepest point. If the estimated dead time remains the same, being determined by the mixer, there will be no need to change the tuning of the controller as flow changes.
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